I would be careful about spinning the speedo, as one of the big problems I've seen is that the odo gears (both the trip and regular) will strip if you power up the speedo on a unit that has set for a long time. What I've discovered by taking several of these apart, is the the fiber gears running the two odos are pressed on a shaft that runs in the aluminum housing. The shafts are greased from the factory, and over time, this grease turns to glue...run the speedo, and you will strip the fiber gears. I went through great pains adapting some nylon gears from McMaster to get my odos to work.
If you take the mechanism out of the casing, getting at the odo shaft and gears is pretty straight forward. Pop them out, clean up the grease that turned to glue and you should be OK. I opted to lubricate the shafts with some oil rather then grease.
Also, as Moses suggests, a small cheap screwdriver will work fine to drive the speedo. The hole is square, so I found one the fit it diagonally and cut the handle off so I could chuck it in the drill.